Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Instabilities in the quantum Sherrington-Kirkpatrick Ising spin glass in transverse and longitudinal fields

T. K. Kopeć, Klaus D. Usadel, G. Büttner

Published Jun 1, 1989• 45 citations
Publisher

Abstract

The infinite-range quantum Ising spin glass in transverse ($\ensuremath{\Gamma}$) and longitudinal ($h$) fields is studied by means of thermo-field-dynamics as a substitute for the $n$-replica trick. With the use of a one-loop approximation for the dynamic self-interaction the stability of the mean-field-type solution against the action of fluctuations is investigated in the parameter space (${k}_{B}T, \ensuremath{\Gamma}, h$).

DOI
10.1103/physrevb.39.12418

Related papers

Stability of the quantum Sherrington-Kirkpatrick spin glass model

A. P. Young

I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).

arXiv:1707.07107•Physical review. E•Sep 11, 2017•29 citations

Quantum Heisenberg spin glasses: Anisotropy effects and field dependence

T. K. Kopeć, G. Büttner, Klaus D. Usadel

The infinite-range quantum Heisenberg spin glass with uniaxial anisotropy (D) and coaxial external magnetic field (h) is studied by means of the thermo-field-dynamics as a substitute for the n-replica trick. Within the mean-field theory a multiplicity of spin-glass phases has been found for the spin values S=1 and S=3/2, respectively, including: longitudinal, transverse, and mixed phases. The stability of the mean-field-type solution against the action of fluctuations has been investigated and the existence of a crossover from the longitudinal-to-transverse freezing is demonstrated explicitly in the temperature-field (T-h) phase diagram for various values of the constant D.

Physical review. B, Condensed matter•May 1, 1990•25 citations

A dynamic theory of transverse freezing in the Sherrington-Kirkpatrick Ising model

T. K. Kopeć

The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the real-time finite-temperature quantum-field theory) due to Takahashi and Umezawa as a substitute for the 'n-replica trick'. An exact effective Lagrangian for the average dynamic causal and response functions is derived by using the functional-integral approach. Furthermore, the properties of the Lagrangian are studied in the mean-field limit by considering the infinite-range random exchange. In this limit the self-consistent equations for the matrix components of the TFD causal Green function are derived with the effective thermal Hamiltonian containing the dynamic spin self-interaction. The static properties of this equation are studied above and below the freezing temperature T c by extracting the time-delocalised parts of dynamic autocorrelations. The self-consistent equations for the spin-glass order parameter and local susceptibility are obtained. The classical ( Gamma to 0) and quantum (T to 0) limits are discussed. The crucial role of the choice of the thermal-state condition for the spin-glass problem and its striking similarity to the non-equilibrium formulation of the TFD are pointed out.

Journal of Physics C Solid State Physics•Dec 30, 1988•31 citations

Transverse freezing in the quantum Ising spin glass: a thermofield dynamic approach

T. K. Kopeć

The effect of a transverse field on the spin-glass freezing transition in the Sherrington-Kirkpatrick version of the quantum Ising model is studied using the thermofield dynamics (the real-time, finite-temperature quantum field theory) due to Takahashi and Umezawa (1975) as a substitute for the 'n-replica trick'. The field-dependent transition line to the spin-glass phase is found and the averaged local static metastable susceptibility in the transverse direction is evaluated in the simple mean-field approximation for the effective dynamic Lagrangian.

Journal of Physics C Solid State Physics•Jan 20, 1988•50 citations

Stability analysis of an Ising spin glass with transverse field

G. Büttner, Klaus D. Usadel

The stability of the replica-symmetric solution of the Sherrington-Kirkpatrick Ising spin glass with transverse field is investigated. We show that the replica-symmetric solution becomes unstable in the whole spin-glass phase.

Physical review. B, Condensed matter•Jan 1, 1990•68 citations

Infinite-range Ising spin glass with a transverse field under the static approximation

Dohyun Kim, Jong-Jean Kim

In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of ${C}_{H}/T$ near the glass transition temperature ${T}_{g},$ associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.

Physical review. B, Condensed matter•Aug 20, 2002•23 citations
The infinite-range quantum Ising spin glass in transverse (\ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ) and longitudinal (hhh) fields is studied by means of thermo-field-dynamics as a substitute for the nnn-replica trick. With the use of a one-loop approximation for the dynamic self-interaction the stability of the mean-field-type solution against the action of fluctuations is investigated in the parameter space (kBT,\ensuremathΓ,h{k}_{B}T, \ensuremath{\Gamma}, hkB​T,\ensuremathΓ,h).
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
The infinite-range quantum Heisenberg spin glass with uniaxial anisotropy (D) and coaxial external magnetic field (h) is studied by means of the thermo-field-dynamics as a substitute for the n-replica trick. Within the mean-field theory a multiplicity of spin-glass phases has been found for the spin values S=1 and S=3/2, respectively, including: longitudinal, transverse, and mixed phases. The stability of the mean-field-type solution against the action of fluctuations has been investigated and the existence of a crossover from the longitudinal-to-transverse freezing is demonstrated explicitly in the temperature-field (T-h) phase diagram for various values of the constant D.
The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the real-time finite-temperature quantum-field theory) due to Takahashi and Umezawa as a substitute for the 'n-replica trick'. An exact effective Lagrangian for the average dynamic causal and response functions is derived by using the functional-integral approach. Furthermore, the properties of the Lagrangian are studied in the mean-field limit by considering the infinite-range random exchange. In this limit the self-consistent equations for the matrix components of the TFD causal Green function are derived with the effective thermal Hamiltonian containing the dynamic spin self-interaction. The static properties of this equation are studied above and below the freezing temperature T c by extracting the time-delocalised parts of dynamic autocorrelations. The self-consistent equations for the spin-glass order parameter and local susceptibility are obtained. The classical ( Gamma to 0) and quantum (T to 0) limits are discussed. The crucial role of the choice of the thermal-state condition for the spin-glass problem and its striking similarity to the non-equilibrium formulation of the TFD are pointed out.
The effect of a transverse field on the spin-glass freezing transition in the Sherrington-Kirkpatrick version of the quantum Ising model is studied using the thermofield dynamics (the real-time, finite-temperature quantum field theory) due to Takahashi and Umezawa (1975) as a substitute for the 'n-replica trick'. The field-dependent transition line to the spin-glass phase is found and the averaged local static metastable susceptibility in the transverse direction is evaluated in the simple mean-field approximation for the effective dynamic Lagrangian.
In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of CH/T{C}_{H}/TCH​/T near the glass transition temperature Tg,{T}_{g},Tg​, associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.